1 Definition
The characteristic polynomial is a basic invariant associated with a square matrix or linear operator. It summarizes how the transformation acts on vectors and is especially important because its roots are the eigenvalues of the transformation. In finite-dimensional linear algebra, it provides a bridge between algebraic formulas and geometric behavior.
1.1 Characteristic polynomial of a matrix
For an \(n \times n\) matrix \(A\), the characteristic polynomial is usually defined by \[ p_A(\lambda) = \det(\lambda I - A), \] where \(I\) is the \(n \times n\) identity matrix and \(\lambda\) is an indeterminate. This produces a polynomial in \(\lambda\) of degree \(n\). The coefficients depend on the entries of \(A\), but the polynomial itself captures properties that remain unchanged under change of basis.
1.2 Characteristic polynomial of a linear operator
If \(T: V \to V\) is a linear operator on an \(n\)-dimensional vector space, its characteristic polynomial is defined by choosing any basis for \(V\), writing \(T\) as a matrix \(A\), and then taking the characteristic polynomial of \(A\). Because similar matrices have the same characteristic polynomial, the result does not depend on the chosen basis.
1.3 Equivalent formulations
The characteristic polynomial can be presented in more than one equivalent way. The determinant formula is the most common, but the polynomial can also be understood through its roots and their relation to eigenvalues.
1.3.1 Determinant definition
The determinant definition directly encodes the failure of \(\lambda I - A\) to be invertible. Values of \(\lambda\) for which the determinant vanishes correspond to eigenvalues, since then \(\lambda I - A\) has a nontrivial kernel.
1.3.2 Eigenvalue-based interpretation
From the eigenvalue perspective, the characteristic polynomial is the polynomial whose zeros are exactly the eigenvalues of the matrix, counted with algebraic multiplicity. This interpretation makes the polynomial a central tool for spectral analysis.
1.4 Notation and conventions
Common notation includes \(p_A(\lambda)\), \(\chi_A(\lambda)\), and sometimes \(c_A(\lambda)\). Most modern texts define it as \(\det(\lambda I - A)\), although some older sources use \(\det(A - \lambda I)\), which differs only by a factor of \((-1)^n\). The choice of convention affects the leading coefficient but not the roots.
2 Fundamental properties
Several structural properties make the characteristic polynomial especially useful. It has fixed degree, reflects basic matrix invariants, and behaves predictably under similarity and certain algebraic transformations.
2.1 Degree and leading coefficient
For an \(n \times n\) matrix, the characteristic polynomial has degree \(n\). Under the convention \(p_A(\lambda)=\det(\lambda I-A)\), the leading coefficient is \(1\), so the polynomial is monic. The constant term is \((-1)^n\det(A)\).
2.2 Relationship to trace and determinant
The coefficient of \(\lambda^{n-1}\) is \(-\operatorname{tr}(A)\) under the standard convention. Thus the trace and determinant appear as the first and last coefficients, respectively. More generally, the intermediate coefficients are related to sums of principal minors and symmetric functions of the eigenvalues.
2.3 Similarity invariance
If \(B = P^{-1}AP\) for an invertible matrix \(P\), then \(A\) and \(B\) are similar and have the same characteristic polynomial. This follows from \[ \det(\lambda I - B)=\det\bigl(P^{-1}(\lambda I - A)P\bigr)=\det(\lambda I-A). \] As a result, the characteristic polynomial depends only on the linear transformation, not on the basis used to represent it.
2.4 Behavior under scalar multiplication and shifts
If \(A\) is replaced by \(cA\) for a scalar \(c\), the characteristic polynomial changes in a predictable way through the substitution \(\lambda I - cA\). Likewise, shifting by a scalar multiple of the identity changes the variable by translation: the characteristic polynomial of \(A + cI\) is obtained from that of \(A\) by replacing \(\lambda\) with \(\lambda - c\). These rules are often useful in simplifying computations.
3 Eigenvalues and roots
The most important connection between the characteristic polynomial and linear algebra is its relationship to eigenvalues. The roots of the polynomial determine the spectral data of the transformation.
3.1 Roots of the characteristic polynomial
A scalar \(\lambda\) is an eigenvalue of \(A\) if and only if \[ \det(\lambda I - A)=0. \] Thus the eigenvalues are exactly the zeros of the characteristic polynomial. This provides an algebraic method for detecting special directions preserved by the matrix.
3.2 Algebraic multiplicity
The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. If \(\lambda_0\) appears \(k\) times as a root, then \(\lambda_0\) has algebraic multiplicity \(k\). The sum of all algebraic multiplicities equals the dimension of the underlying space.
3.3 Geometric multiplicity
The geometric multiplicity of an eigenvalue is the dimension of its eigenspace. It is always at least 1 for an eigenvalue and never exceeds the algebraic multiplicity. When geometric multiplicity equals algebraic multiplicity for every eigenvalue, the matrix is especially well behaved and may be diagonalizable.
3.4 Spectral interpretation
The characteristic polynomial encodes the spectrum of the matrix, meaning the complete set of eigenvalues. For many classes of matrices, especially over the complex numbers, this spectrum governs long-term behavior, decomposition into invariant subspaces, and canonical forms. In this sense, the polynomial serves as a compact algebraic summary of spectral structure.
4 Computation
Although the characteristic polynomial has a clean definition, explicit calculation can be challenging for large matrices. Several techniques are used in practice, ranging from direct expansion to exploiting special matrix structure.
4.1 Direct determinant expansion
The most direct method is to compute \(\det(\lambda I-A)\) by expanding the determinant symbolically. This works well for small matrices, especially \(2 \times 2\) and \(3 \times 3\) cases, but becomes cumbersome as dimension increases because the expressions grow rapidly.
4.2 Use of row and column operations
Elementary row and column operations can simplify the determinant, provided one keeps track of how each operation affects the value. Since the matrix involves a polynomial variable \(\lambda\), such operations are often arranged to reduce the matrix toward triangular form or to isolate blocks.
4.3 Special matrices
Certain matrix families have characteristic polynomials that can be written down immediately or with little effort.
4.3.1 Diagonal matrices
If \(A\) is diagonal with entries \(d_1,\dots,d_n\), then \[ p_A(\lambda)=\prod_{i=1}^n (\lambda-d_i). \] The eigenvalues are exactly the diagonal entries, counted with multiplicity.
4.3.2 Triangular matrices
For upper or lower triangular matrices, the characteristic polynomial is again \[ p_A(\lambda)=\prod_{i=1}^n (\lambda-a_{ii}), \] where \(a_{ii}\) are the diagonal entries. The off-diagonal entries do not affect the characteristic polynomial, though they may affect eigenvectors and diagonalizability.
4.3.3 Block diagonal matrices
If a matrix is block diagonal, its characteristic polynomial is the product of the characteristic polynomials of the blocks. This allows large problems to be broken into smaller ones and is especially useful when a matrix decomposes into independent components.
4.4 Companion matrices
A companion matrix is constructed from a monic polynomial so that the polynomial is its characteristic polynomial. This provides a canonical way to realize any monic polynomial as the characteristic polynomial of some matrix. Companion matrices are important in theory and in algorithms for polynomial representation.
5 Theorems and applications
The characteristic polynomial appears in several fundamental theorems and practical methods. It links eigenvalues to matrix identities, canonical forms, and criteria for simplifying linear transformations.
5.1 Cayley–Hamilton theorem
The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic polynomial: \[ p_A(A)=0. \] This means that if \(p_A(\lambda)\) is substituted by the matrix \(A\), the resulting matrix expression vanishes. The theorem is central to polynomial identities involving matrices and has many consequences, including formulas for powers of matrices and inverses when they exist.
5.2 Diagonalization criteria
The characteristic polynomial helps determine whether a matrix can be diagonalized. If the polynomial splits into distinct linear factors and the eigenspaces have the expected dimensions, then the matrix is diagonalizable. More generally, repeated roots indicate that additional analysis is needed to compare algebraic and geometric multiplicities.
5.3 Jordan normal form
Over an algebraically closed field, the Jordan normal form refines the information in the characteristic polynomial by organizing eigenvalues into blocks. The polynomial gives the eigenvalues and their multiplicities, while the Jordan form reveals the sizes of the blocks and therefore the nilpotent structure around each eigenvalue.
5.4 Minimal polynomial comparison
The characteristic polynomial should be distinguished from the minimal polynomial. Both are annihilating polynomials, but the minimal polynomial is the monic polynomial of smallest degree that annihilates the matrix. The characteristic polynomial is typically larger and contains more multiplicity information, while the minimal polynomial more directly controls the matrix’s algebraic behavior.
6 Special cases
In particular classes of matrices, the characteristic polynomial has additional structure. These cases are often easier to analyze and reveal deeper connections with geometry and operator theory.
6.1 1×1 and 2×2 matrices
For a \(1 \times 1\) matrix \([a]\), the characteristic polynomial is simply \(\lambda-a\). For a \(2 \times 2\) matrix \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \] the characteristic polynomial is \[ \lambda^2 - (a+d)\lambda + (ad-bc). \] This formula highlights the roles of trace and determinant in low dimensions.
6.2 Symmetric and Hermitian matrices
For real symmetric matrices and complex Hermitian matrices, all eigenvalues are real. Consequently, their characteristic polynomials have only real roots. These matrices are especially important because they admit orthogonal or unitary diagonalization, which makes the spectral data easy to interpret.
6.3 Orthogonal and unitary matrices
Orthogonal and unitary matrices preserve length and inner products. Their eigenvalues lie on the unit circle in the complex plane. The characteristic polynomial therefore has roots of absolute value 1, and the polynomial reflects the rotational or reflection-like behavior of these transformations.
6.4 Nilpotent matrices
If \(A\) is nilpotent, then some power of \(A\) is zero. In that case, all eigenvalues are \(0\), and the characteristic polynomial has the form \(\lambda^n\) for an \(n \times n\) nilpotent matrix. This makes nilpotent matrices a useful test case for understanding repeated roots and Jordan structure.
7 Related concepts
The characteristic polynomial is closely connected to several other ideas in linear algebra and applied mathematics. These related notions often provide complementary information about a matrix or operator.
7.1 Minimal polynomial
The minimal polynomial is the smallest monic polynomial that annihilates a matrix. It divides the characteristic polynomial and often gives a more precise description of the matrix’s canonical form. Together, the two polynomials are used to analyze diagonalizability and Jordan structure.
7.2 Characteristic equation
The characteristic equation is the equation \[ \det(\lambda I-A)=0. \] Its solutions are the eigenvalues of \(A\). In practice, the term is often used informally to mean the equation obtained from the characteristic polynomial by setting it equal to zero.
7.3 Spectral polynomial methods
Spectral polynomial methods use polynomials in a matrix or operator to study its spectrum and dynamics. They appear in numerical analysis, iterative algorithms, and systems theory. The characteristic polynomial is one of the foundational examples in this broader family of techniques.
7.4 Invariants under similarity
Similarity invariants are quantities preserved when a matrix is changed by a basis transformation. The characteristic polynomial is one of the most important such invariants, alongside the trace, determinant, rank-related data, and canonical forms. These invariants help classify linear transformations up to equivalence.